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PBEFACE
known men -who helped to prepare the way for the vast develop-
ment of mathematics in the nineteenth and twentieth centuries.
As we pass from the eighteenth to the nineteenth century
we are overwhelmed by a tidal wave of free inventiveness.
New departments of mathematics were created and developed
hi bewildering profusion. The great mathematicians of the nine-
teenth century, some of whom are presented .here, seem
to be almost of a different species from their predecessors. The
new men were not content with special problems, but attacked
and solved general problems whose solutions yielded those of a
multitude of problems which, in the eighteenth century, would
have been considered one by one. A striking example has often
been noted in the contrast between Gauss (1777-1855) and Abel
(1802-29) in the theory of algebraic equations. There is a
similar distinction in the matter of geometry between Gauss
and his pupil Riemann (1826-66). It is no disparagement of
Gauss, but merely a statement of historical fact, to say that he
was content with the problem of finding the algebraic solution
of binomial equations, and did not even mention the general
problem, solved by Abel and Galois (1811-32), of determining
necessary and sufficient conditions that any given algebraic
equation be solvable by radicals. The nature of the general
problem is explained in the accounts given here of Abel and
Galois. Of course there is a certain loose continuity in all
mathematics, clear back to Babylon and Egypt, but the
interesting and fruitful points on the curve of progress are the
dJbc^ntinuities that appear when the curve is closely analysed
as in that just noted of Gauss, Abel, and Galois. One such from
the 1930*s must suffice here as a current example.
The paradoxes of Zeno and the repeated attempts to establish
the differential and integral calculus on a firm logical foundation
exercised mathematicians as early as the seventeenth century,
and continued to worry them all through the second hfrlf of the
nineteenth. Among those who struggled at the task were three
whom we shall meet later. Cantor, Dedekind, and Weierstrass.
Deddkind admitted failure. But failure or not. to achieve the
desired end, the work of all three gave a tremendous impulse
to the study of all mathematical reasoning. How was it to be